# Evans' PDE Exercise 6.6: Weak solution of Dirichlet-Neumann boundary value problem

The exercise is the exercise 6.6 from Evans' PDE.

Suppose $$U$$ is connected, and $$\partial U$$ consists of two disjoint, closed sets $$\Gamma_1$$ and $$\Gamma_2$$. Define what it means for $$u$$ to be a weak solution of Poisson equation with mixed Dirichlet-Neumann boundary conditions: $$\begin{cases} -\Delta u = f \ \ \ \text{in U} \\ u = 0 \ \ \ \text{on \Gamma_1} \\ \frac{\partial u}{\partial \nu} = 0 \ \ \text{on \Gamma_2}. \end{cases}$$

My attempts: Let $$u \in C^{\infty}(U)$$ be a solution to the above problem. Then for $$v \in H^1(U)$$, integration by parts yields \begin{align} (f,v) = -\int_U (\Delta u) v & = \int_U Du \cdot Dv -\int_{\partial U} \frac{\partial u}{\partial \nu} v \\ & = \int_U Du\cdot Dv - \int_{\Gamma_1} \frac{\partial u}{\partial \nu} v \end{align}

I wish to conclude $$\int_{\Gamma_1} \frac{\partial u}{\partial \nu} v = 0$$, but I don't know how to do that. Could anyone give me some hint?

You can pick the space of test functions $$v$$ to be the space $$H^1_{\Gamma_1}(U) = \{v \in H^1(U)\colon v = 0 \text{ on } \Gamma_1 \},$$ which is a Hilbert space with respect to the inner product in $$H^1$$. Actually, you can verify that in $$H^1_{\Gamma_1}$$ the norm $$\|v\|_{H^1_{\Gamma_1}(U)} = \|Dv\|_{L^2(U)}$$ is equivalent to the $$H^1$$ norm (by Poincaré inequality). Then the second integral equals $$0$$ since $$v \in H^1_{\Gamma_1}(U).$$
• Thank you! May I ask that: what's the explicit definition of your "scalar product in $H_1$"? According to the context, it seems that you're not referring to the $H^1$ inner product. But I was thinking if we could use $H^1$ inner product directly. It suffices to check the completeness. Since $H^1$ is already complete, suffices to check closeness. If $u_n \in H^1_{\Gamma_1} \to u$ in $H^1$, then by trace theorem, $u_n \to u$ in $L^2(\Gamma_1)$. Then certainly there has a subsequence $u_n \to u$ a.e. in $\Gamma_1$. That is, $u = 0$ at $\Gamma_1$, saying that $u \in H^1_{\Gamma_1}$. Am I correct? Jul 16 '20 at 14:12